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Computer‐algebra multiple‐timescale method for spatially periodic thin‐film viscous‐flow problems
Author(s) -
Groh Christian M.,
Kelmanson Mark A.
Publication year - 2010
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.2458
Subject(s) - benchmark (surveying) , symbolic computation , mathematics , free surface , cylinder , flow (mathematics) , ordinary differential equation , partial differential equation , mathematical analysis , geometry , mechanics , physics , differential equation , geodesy , geography
SUMMARY A flexible, fully automated, computer‐algebra algorithm is developed for solving a class of non‐linear partial‐differential evolution equations arising frequently in the modeling of two‐dimensional transient free‐surface viscous thin‐film flows. The method, which is formulated for solving spatially periodic problems, is based upon an explicit multiple‐timescale asymptotic approximation of the thin‐film thickness. It admits the resolution of diverse physical phenomena by employing a finite geometric progression of increasingly slow timescales. The method is implemented on a challenging test problem comprising the evolution of an annular film of viscous liquid, with a free surface, adhering to the exterior of a horizontal rotating circular cylinder; as a model for numerous industrially motivated coating flows , this benchmark problem has been analyzed in diverse numerical and theoretical studies, against whose results those of the present method are compared. The explicit algebraic form of the solution admits a study of large‐time evolutionary dynamics that lies beyond the reach of considerably more expensive conventional numerical solvers, thereby shedding new light on the hitherto‐undiscovered explicit dependence of large‐time evolutionary fluid dynamics in terms of independent parameters describing gravitational and capillary effects. The results obtained from the new computer‐algebra procedure are demonstrated to be in good agreement with those obtained from a bespoke efficient numerical integration method that is spectrally accurate in space and 8th‐order (Runge–Kutta) in time. Newly discovered mechanisms describing the decay of free‐surface wave modes, from arbitrary initial conditions to the steady state, are presented. Copyright © 2010 John Wiley & Sons, Ltd.